Article
Comparison of Bayesian Methods on Parameter
Identification for a Viscoplastic Model with Damage
Ehsan Adeli1,*, Bojana Rosi´c2, Hermann G. Matthies3, Sven Reinstädler4and Dieter Dinkler4
1 Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112, USA 2 Applied Mechanics and Data Analysis, University of Twente, 7522 NB Enschede, The Netherlands;
wire@tu-bs.de
3 Institute of Scientific Computing, Technische Universität Braunschweig, 38106 Braunschweig, Germany; h.matthies@tu-bs.de
4 Institute of Structural Analysis, Technische Universität Braunschweig, 38106 Braunschweig, Germany; statik@tu-braunschweig.de (S.R.); d.dinkler@tu-braunschweig.de (D.D.)
* Correspondence: e.adeli@sci.utah.edu
Received: 24 May 2020; Accepted: 24 June 2020; Published: 1 July 2020
Abstract:The state of materials and accordingly the properties of structures are changing over the
period of use, which may influence the reliability and quality of the structure during its life-time. Therefore, identification of the model parameters of the system is a topic which has attracted attention in the content of structural health monitoring. The parameters of a constitutive model are usually identified by minimization of the difference between model response and experimental data. However, the measurement errors and differences in the specimens lead to deviations in the determined parameters. In this article, the focus is on the identification of material parameters of a viscoplastic damaging material using a stochastic simulation technique to generate artificial data which exhibit the same stochastic behavior as experimental data. It is proposed to use Bayesian inverse methods for parameter identification and therefore the model and damage parameters are identified by applying the Transitional Markov Chain Monte Carlo Method (TMCMC) and Gauss-Markov-Kalman filter (GMKF) approach. Identified parameters by using these two Bayesian approaches are compared with the true parameters in the simulation and with each other, and the efficiency of the identification methods is discussed. The aim of this study is to observe which one of the mentioned methods is more suitable and efficient to identify the model and damage parameters of a material model, as a highly non-linear model, using a limited surface displacement measurement vector and see how much information is indeed needed to estimate the parameters accurately.
Keywords:viscoplastic-damage model; uncertainty quantification; Bayesian parameter and damage
identification; functional approximation
1. Introduction
In order to predict the behavior of mechanically loaded metallic materials, constitutive models are applied, which present a mathematical frame for the description of elastic and inelastic deformation. All inelastic constitutive models contain parameters which have to be identified for a given material from experiments [1].
There are few investigations on the simplest material model, the elasticity model, to identify only very few parameters of the model where sampling approaches like the Metropolis algorithm and its modifications are employed. Pacheco et al. [2] investigated a three-point bending test to identify elastic behavior and calibration is done by solving the inverse problem in a Bayesian setting by using the Metropolis-Hastings algorithm. Only elastic moduli are identified without considering
the error. Slonski et al. [3] also applied a sequential particle filter on an elastic model and Young’s modulus was estimated, where the Bayesian setting was compared to the deterministic approach, and the Bayesian setting was preferred. The elastic modulus of a polymeric material is updated by Zhang et al. [4], where a Markov Chain Monte Carlo method is used, but very high computation time is reported. Further, Young’s modulus is estimated for a considered material by Gallina et al. [5] by applying a multi dimensional Markov Chain Monte Carlo method. A similar Bayesian approach for composite materials to estimate Young’s modulus is carried out by Pieczonka et al. [6]. Arnst et al. [7] have applied a Markov Chain Monte Carlo method by using polynomial chaos expansion to identify Young’s modulus of an elastic model.
Only few investigations were carried out on enriched material models such as a viscoelastic model to identify the few parameters by employing the Metropolis-Hastings technique and the classical Markov Chain Monte Carlo method. Rappel et al. [8] studied an elastic and a viscoelastic model where the measurement error is considered. Bayesian inference is applied to estimate only the elasticity modulus by applying an adaptive Metropolis-Hastings technique. An et al. [9] investigated a crack model by a classical Markov Chain Monte Carlo method in order to estimate the parameters of a model which represent the size and position of the crack in the Bayesian setting. Also Hernandez et al. [10] applied a Markov Chain Monte Carlo method on a viscoelastic model in order to update its model parameters in a Bayesian setting, but the posterior distributions of the parameters are not updated properly. In fact, the parameters are not identified properly. Mahnken [11] has also applied a Markov Chain Monte Carlo method to estimate few parameters of a plasticity model. The damage parameters of a truss structure under model uncertainties are studied by Zheng et al. [12], where a multi-level Markov Chain Monte Carlo method is applied and the true values are not well estimated. Further, this approach suffers from high computation time. Another damage detection approach is applied by Nichols et al. [13] by applying a modified version of the Markov Chain Monte Carlo method. There are other investigations in the Bayesian setting using the Markov Chain Monte Carlo method, and Madireddy et al [14], Wang and Zabaras [15], and Oh et al. [16] have carried out investigations on the identification of a material model by using its modified method. In studies carried out by Alvin [17], Marwala and Sibusiso [18], Daghia et al. [19], Abhinav and Manohar [20], Gogu et al. [21,22], and Koutsourelakis [23,24], the elastic parameters of the model are estimated stochastically. Fitzenz et al. [25], Most [26], and Sarkar et al. [27] investigated elastoplastic materials and thermodynamical material models to identify their model parameters. Other studies on viscoelastic models are carried out by Zhang et al. [28], Mehrez et al. [29], and Miles et al. [30]. Further investigations on viscoelastic models to estimate a higher number of model parameters are studied by Zhao and Pelegri [31], and by Kenz et al. [32]. The estimation of fatigue parameters using Markov Chain Monte Carlo methods is also studied by An et al. [33].
Few investigations in which the Kalman filter and its modifications are employed to identify the material model parameters can be found in literature. Hoshi et al. [34] have estimated the Young’s Modulus and Poisson’s ratio parameters of an organ model by an extended version of Kalman filter. The same study on an elastic model is carried out to identify the elastic constants of anisotropic materials using Kalman filter by Furukawa et al. [35]. Conte et al. [36] applied some modifications of Kalman filter on a nonlinear structural model to identify its parameters. Hendriks [37] investigated the possibility of identification of few parameters on a viscoelastic model representing the behavior of solid materials by using Kalman filter. Few parameters to identify on the crack path is carried out by Bolzon et al. [38] by employing Kalman filter. Nguyen et al. and Mahmoudi et al. [39–41] applied an extended Kalman filter on an elastoplastic model to identify the model parameters. Wall [42] and Nakamura et al. [43] also applied a modification of Kalman filter on a viscoplastic model to determine its model parameters probabilistically. Further, Agmell et al. [44] have determined the Johnson–Cook constitutive model constants for an orthogonal cutting process by using the Kalman filter. Sevieri et al. [45–47] investigated the parameter identification of structural models
in the framework of structural health monitoring. Marsili et al. [48–52] investigated the update the finite element model using functional approximation of the system response.
Elastic–plastic graded materials are observed by Bocciarelli et al. [53] and Gu et al. [54] in order to identify their model parameters by employing Kalman filter. The parameters of an elastic-damage interface model are also identified by Corigliano et al. [55,56] where the extended Kalman filter method is applied. Further, Ebrahimian et al. and Hariri-Ardebili et al. [57–62] investigated the damage parameter identification in the framework of structural health monitoring by using an extended version of Kalman filter. Damage detection for the purpose of health monitoring is also done by Yan et al. [63] by using the Kalman filter and other stochastic approaches by Kourehli et al. and Gharehbaghi et al. [64–66].
Although these discussed research for the identification of mechanical material models has been carried out, most of these previous research did not identify many parameters, e.g., hardening and damage parameters of the complex mechanical material models, did not quantify mostly the modeling uncertainties, did not properly deal with the ill-conditioning inherent based on the available data, and the methods employed have a high computation time even for the simplest material model considering a very few uncertain parameters. However, the uncertainty associated with the material model predictions can have a significant impact on the decision-making process in design, control, and health monitoring process.
In this paper, a viscoplastic model of Chaboche with damage is studied. The model contains five material parameters and three damage parameters which have to be determined from experimental data by using two Bayesian approaches, i.e., Transitional Markov Chain Monte Carlo method [67,68] and Gauss-Markov-Kalman filter approach [69–72]. The results of these two methods are compared and different aspects of them are evaluated. It should be noted that virtual data is employed instead of real experimental data. In addition, a cyclic tension-compression test is applied in order to extract the virtual data. A cyclic test is employed since the hardening equations as well as the damage equation could get involved so that the information from these equations can be observed.
Sections3and4explain how to propagate the uncertainty in the model. The probabilistic model is reformulated from the deterministic model, and once the forward model is provided, the model parameters are updated using two mentioned stochastic approaches.
In Section5the desired parameters are identified from the measured data and the efficiency of methods is studied. In fact, the parameters which have been considered as uncertain parameters are updated and their uncertainties are narrowed using Bayesian techniques. The results are thoroughly studied and the identified parameters as well as the corresponding model responses are analyzed. Finally the prediction of the models is then compared with the measured data.
2. Model Problem
For viscoplasticity, the dissipation potential is given by
φvp(σ˜) = k n+1
D˜σex k
En+1
with ˜σex= ˜σeq−σy−R and h·i =max(0, x) (1)
in terms of the effective equivalent stress ˜σeqand the isotropic hardening R as time-varying variables.
On the contrary, the yield stress σy, as well as k and n are further model parameters constant in time.
Based on the von-Mises yield-criterion the effective equivalent stress
˜σeq=
q
3 I2(σ˜ −χ)d+D¯(I1(σ˜ −χ))2 (2) is described by the second Invariant
I2(σ˜−χ)d= 1 2tr (σ˜ −χ)d· (σ˜ −χ)d (3)
applied to the deviatoric part of the effective stress tensor ˜σe f =σ˜ −χincorporating the kinematic hardening tensor χ. The second term in Equation (2) takes into account that the elastic capacity is reduced when the material is damaged. Note that the first invariant
I1(σ˜−χ) =1
3tr(σ˜ −χ) (4)
is equivalent to the hydrostatic stress. The partial derivative of the dissipation potential φvp with
respect to ˜σleads to the equation for the effective viscoplastic strain
˙˜evp= ∂φvp ∂ ˜σ = D˜σex k En∂ ˜σex ∂ ˜σ (5)
given in rate formulation. More details are given in AppendixA. The Chaboche model allows for isotropic and kinematic hardening, which is considered in order to describe the Bauschinger effect observed at steel material under high cyclic loadings, see Simo and Hughes [73]. The evolution of the isotropic and kinematic hardening is described by the ordinary differential equations
˙R=bR(HR−R) ˙p and (6a) ˙χ=bχ 2 3Hχ ∂ ˜σeq ∂ ˜σ −χ ˙p (6b)
respectively, depending on the rate of accumulated equivalent viscoplastic strain
˙p=D˜σex
k En
, (7)
using the McAuley bracket. The parameter bRaffects the speed of evolution, whereas the value of
the parameter HRis the threshold for the isotropic hardening. With the same influence and meaning,
the parameters bχand Hχcontrol the kinematic hardening.
The evolution equation for the local damage proposed by Kowalsky et al. [74] reads:
˙ D=c1+c2e−c3p + ˙p++c4(c5−D¯) * tr ∂ ˜σeq ∂ ˜σ ˙p + !+ . (8)
It is regularized by the model parameter c1up to c5and only activated if the threshold eeqvpfor
the viscoplastic strain is reached, thusp2/3 evp : evp >evpeq. Achieving a model where damage only
accumulates with positive hydrostatic stress, Equation (8) is formulated by the active accumulated plastic strain p+, an internal variable increasing with
˙p+= ˙pD I1
|I1|
E
, (9)
see Pirondi and Bon [75]. For the parameter identification, the initial conditions R(0) =0, χ(0) =0
and D(0) =0 are assumed. The complete material model is summarized in Table1.
The governing equations above are numerically solved using the space-time finite element method (ST-FEM), see Hughes and Hulbert [76]. The ST-FEM is the consistent extension of the finite element method [77] with a time integration scheme following the Galerkin method, thus both, in space and time, the differential equations of the structure are approximately solved with the same numerical method.
By gathering all the desired material parameters to identify into the vector
q= [κ G bRbχσyc1c2c3], where κ and G are bulk modulus and shear modulus respectively,
the goal is to estimate q given measurement displacement data, i.e.,
u=Y(q) +ε (10)
in which Y(q)represents the measurement operator and ε the measurement (also possibly the model) error. Being an ill-posed problem, the estimation of q given u is not an easy task and requires regularization. This can be achieved either in a deterministic or probabilistic setting. Here, the latter one is taken into consideration as further described in the text.
Table 1.Chaboche-type material with hardening and damage. Elastic strains ˙˜eel(t) =C−1: ˙˜σ(t) with C(G, κ) Viscoplastic strains ˙˜evp(t) = ˜σ eq(t) −σy−R(t) k n ∂ ˜σex ∂ ˜σ
Isotropic and kinematic hardening
˙R=bR(HR−R) ˙p ˙χ=bχ 2 3Hχ ∂ ˜σeq ∂ ˜σ −χ ˙p Local damage D=D¯ −l2c∇2D¯ ˙ D=c1+c2e−c3p + ˙p++c 4(c5−D¯) * tr ∂ ˜σeq ∂ ˜σ ˙p + !+ Initial conditions
˜eel(0) =0, ˜evp(0) =0, R(0) =0, χ(0) =0and D(0) =0 Parameters and their units
G[N/mm2], κ[N/mm2], (elastic strains)
σy[N/mm2], k[N/mm2], n[−], (viscoplastic strains) bR[−], HR[N/mm2], bχ[−], Hχ[N/mm2], (hardening)
c1[−], c2[−], c3[−], c4[−], c5[−] (local damage)
3. Bayesian Identification
The main methods are applied on the case study and their efficiencies will be studied in Section5. The first method is the so-called Transitional Markov chain Monte Carlo method and the other one is the Gauss-Markov-Kalman filter which are presented in the following.
3.1. Transitional Markov Chain Monte Carlo Method
The Transitional Markov Chain Monte Carlo (TMCMC) method is proposed by Ching and Chen [67,68]. The generation of samples from the posterior PDF can be a difficult task due to the lack of information about the geometry of the probability density function and in order to overcome this problem, the TMCMC algorithm uses a sequence of intermediate PDFs that converge to the target posterior PDF, which are defined by:
where j =0, 1, 2, ..., M and 0= r0 ≤r1 ≤r2 ≤... ≤ rM = 1 in which the index j denotes the stage
number. As one can see, this construction fulfills the desirable properties: the series of intermediate PDFs starts from the prior PDF, i.e., rj =0, and ends with f(θ|M,D), i.e., rj =1, by considering that
f0(θ) = f(θ|M)and fM(θ) = f(θ|M,D), which means:
rj =0 f0(θ)∝ f(θ|M) (12)
rj=1 fM(θ)∝ f(θ|M,D) (13)
The idea is that although the geometry change from f(θ|M)to f(θ|M,D)can be dramatic, the change between two adjacent intermediate PDFs can be small. This small transition makes it possible to obtain efficiently samples from fj+1(θ)based on samples from fj(θ).
Transitional MCMC Algorithm
The idea behind TMCMC is described in detail in this subsection. The entire algorithm is presented in the following [67,68,78].
1. In the first step, j=0 is set and from f0(θ) = f(θ|M)a set of N samples are picked up. 2. rj+1 should be selected in a way that the coefficient of variation of f(D|M, θ(j)k )
rj+1−rj where
k=1, 2, ..., N is equal to a prescribed value. Accordingly, the coefficient of variation serves as an indicator of how close is fj(θ)to fj+1(θ).
3. The plausibility weight w(θ(j)k ) = f(D|M, θk(j))rj+1−rj is calculated for k = 1, 2, ..., N and the parameter Sj=∑N
k=1w(θ (j)
k )/Njis computed.
4. Samples θj+1k for k = 1, 2, ..., N are provided from f(θ)j+1 by Metropolis-Hastings algorithm. In other words, the k-th sample is picked up from a Markov Chain that starts from one of the samples θji where i = 1, 2, ..., N which is selected randomly. The i-th initial sample θ(j)i is chosen with probability w(θ(j)i )/∑Nl=1w(θ(j)l ). As proposal distribution, a Gaussian which is centered at the current sample in the k-th chain is applied in Metropolis-Hastings algorithm. Also, the covariance matrix:
Σj=β2 ∑N i=1w(θ (j) i ) h θ(j)i −µj i h θ(j)i −µj iT ∑N k=1w(θ (j) k ) (14) where µj= ∑ N l=1w(θ (j) l )θ (j) l ∑N l=1w(θ (j) l ) . (15)
in which β is a prescribed scaling factor which is used as the rejection rate in Metropolis-Hastings algorithm.
5. Steps two to four are repeated until rM =1. At the final step, samples θ(M)k for k= 1, 2, ..., N
are distributed according to f(θ|M,D)and it is turned out that S =ΠMj=0Sjis asymptotically
unbiased for f(D|M).
One of the significant advantages of modified Metropolis-Hastings, which is applied in TMCMC, is that the first levels allow a free exploring of the sample space, while, in the last simulation levels, the sampling is performed from a narrower neighborhood of the sample space. Furthermore, the proposal distribution changes within the same simulation level and this property causes a better local behavior. This is accomplished by modifying the proposal distribution for each level, in such a
manner that its standard deviation is small for higher simulation levels, while the average value drives the sampling towards the most important neighborhood of the sample space [67,68,78].
3.2. Gauss-Markov-Kalman Filter Using Functional Approximation
Some methods used for estimation of Bayes’s theorem unlike Monte Carlo methods do not use all information but only part of information as approximations are considered. Hence the balance between time consumption, considered amount of information and the accuracy of approximations can be achieved. Incidentally, this leads to Kalman filter (KF) [79–81] method as it was related to Gauss-Markov theorem which is developed without any reference to Bayes’s theorem. Moreover, the polynomial chaos expansion is also used along with the ensemble Kalman filter (EnKF) [69,82] in order to be completely independent from any time consuming computational implementations such as Monte Carlo method. This leads to the Gauss-Markov-Kalman filter [69–72] which is discussed in this section and more details are provided in AppendixB.
3.2.1. The Linear Filter
The minimization problem as in the Equation (A26) is to be solved and it lies in infinite dimensional space. Hence it is to be approximated using Galerkin method in finite dimensional subspaces. The chosen desired subspace to solve the problem isQ1⊂Q∞⊂Q. Hence the desired
subspace is shown in Equation (16) where the affine maps Φ are certainly measurable [69–71,83]. Q1= {z : z=Φ(y) =L(y(ω)) +b, L∈L(Y,Q), b∈ Q} ⊂Q∞⊂Q (16)
The Equation (17) is obtained from the minimization problem as given by the Equation (A26) where the optimal affine map is introduced via so-called Kalman gain K∈L(Y,Q). The Kalman gain is represented as K := cov(q, y)cov(y)−1where cov(q, y)is the covariance of q and y, cov(y)is the auto-covariance of y and a∈ Qread as a := ¯q−K(¯y).
||q− (K(y) +a)||2Q=min
L,b ||q− (L(y) +b)|| 2
Q (17)
It should be noted that asQ1 ⊂ Q∞ is a true subspace, then obviously some information is
disregarded when using this approximation g(y) = K(y) +a. Although the computation becomes easier, some information that we may learn from the measurement is neglected. Equation (18) is determined from the Equation (A23) and from the described algorithm.
qa,1L=qf+ (K(ˆy) −K(y)) =qf +K(ˆy−y) (18)
This linear filter is called Gauss-Markov-Kalman filter (GMKF) with the linear minimum mean square error K(ˆy)defined as in [69–72]. It should be noted that Gauss-Markov-Kalman filter is a general form of the original Kalman filter which is considered only for the mean values of the random variables of the parameters included in Equation (18). Accordingly the Equation (A25) representing the algorithm turns to the Equation (19).
qn+1=qn+K((ˆyn+1) − (Y(qn) +e)) =qn−K(Y(qn) +e)) +K(ˆyn+1) (19)
Equation (20) is determined from the Equation (18) by introducing the Kalman gain and by considering the random variables as the argument. By considering the error the Kalman gain is defined as K :=cov(qf, y)(cov(y) +cov(e))†.
The Gauss-Markov-Kalman filter as described in Equation (20) needs to be discretized in order to implement numerically as it is related with the random variables.
3.2.2. Sequential Gauss-Markov-Kalman Filter
The process of updating can be done several times on the whole time interval. Once a high non-linear chaotic system is divided to very small time steps, the problem turns to plenty of continuous linear systems so that the GMKF approach can update the model parameters in a much better way if it is applied several times to update the model parameters on each time step. Eventually this approach helps to update the uncertain parameters of a non-linear system. Therefore the Gauss-Markov-Kalman filter as in Equation (20) can be written as like in Equation (21) on the k-th time step for the n total time steps, where the Kalman gain reads K(k):=cov(q(k)f , y(k))(cov(y(k)) +cov(e))†.
q(k)a (ω) =q(k−1)a (ω) +K(k)(ˆy(k)−y(k)(ω)) (21)
The Equation (21) is called sequential Gauss-Markov-Kalman filter. Schematically the process of sequential Gauss-Markov-Kalman filter is shown in Figure1where the posterior of one update is the prior of the next update.
Model at Model at Modelat t1 t2 tn 0 a q 1 a q n1 a q
...
n a q 1 y 2 y n yFigure 1. Sequential GMKF method.
The approaches such as sampling and spectral approximation or functional approximations to compute the linear filter approximations are discussed in this section.
3.2.3. Sampling
Considering N random variables an ensemble of sampling points ω= [ω1, ..., ωN]are taken into
account [82]. The Gauss-Markov-Kalman filter as shown in Equation (20) for the considered ensemble of sampling points results in Equation (22).
∀` =1, ..., N : qa(ω`) =qf(ω`) +Cqfy(Cy+Ce)†(ˇy−y(ω`)) (22)
The Equation (22) is the basis of ensemble Gauss-Markov-Kalman filter where Cqfy=cov(qf, y), Cy=cov(y)and Ce=cov(e). qf(ωl)and qa(ωl)are shown as particles in the extended version.
3.2.4. Functional Approximation
The discretization of the random variables are performed by spectral or functional approximations instead of sampling [70,84,85]. Hence the desired random variables are described as functions of known random variables{θ1(ω), ..., θl(ω), ...}. As only finite random variables can be dealt, a finite vector
random variables in functional representation θ(ω) = [θ1(ω), ..., θn(ω)]can be considered where n
random variables θ are taken into account.
The polynomial chaos expansion [86,87] is chosen as system of functions but also other possibilities exist. It should be noted that the finite set of linear independent Hermite functions{Hα}α∈JM of
expansion [69–71]. The multi-index is represented by α and the setJMis a finite set with cardinality M.
The functional approximation of a random variable q(ω)is shown in Equation (23).
q(ω) =
∑
α∈JM
qαHα(θ(ω)) =
∑
α∈JMqαHα(θ) =q(θ) (23)
The argument ω is neglected in the Equation (23) because the probability measurePonΩ is
transported to Θ=Θ1×...×Θn. The range of θ showingPθ = P1×...× Pn as a product measure, whereP`= (θ`)∗Pis the distribution measure of the random variable θ`and it is noted that the random
variables θ`are independent. Therefore all computations are performed on Θ which is typically a
subset ofRn. Hence the Gauss-Markov-Kalman filter as shown in Equation (20) for the considered
expansion results in Equation (24), with the indicated inverse sign †, which is known as spectral Gauss-Markov-Kalman filter.
qa(θ) =qf(θ) +Cqfy(Cy+Ce)†(ˇy−y(θ)) =qf(θ) +K(ˇy−y(θ)) (24) It should also be noted that in spectral approximation the Gauss-Markov-Kalman filter as shown in the Equation (20) has the same form as of the sampling approach but the only difference is the functional approximation of random variables, i.e., the Hermite functions are used to calculate the covariance matrices. For instance Cqfycan be easily computed as given in Equation (25).
Cqfy=
∑
α>0
α!(q(α)f (θ))(y(α)(θ))T (25) While the Equation (20) is applied on samples or particles in sampling approach, in spectral approximations it is applied on the coefficients as shown in Equation (23) which are the functional approximation of the random variables.
4. Numerical Results
The identification of the material constants in the Chaboche unified viscoplasticity model with damage is a reverse process based on virtual data. The aim of the parameter estimation is to find a parameter vector q introduced in the previous section. The bulk modulus (κ), the shear modulus (G), the isotropic hardening coefficient (bR), the kinematic hardening coefficient (bχ) and the yield stress (σy) as well as damage parameters (c1, c2 and c3) are considered as uncertain parameters of the
constitutive model.
As both kinds of hardenings are taken into account and the studied model is a damaged material model, one appropriate way of the identification of parameters is to use the results of the cyclic tests as observation, since more information can be obtained from virtual data rather than from creep and relaxation tests, especially information regarding hardening and damage parameters as the hardening and damage equations are involved in this case.
Two tests are studied to observe the efficiency of the identified parameters using different Bayesian approaches which are discussed in the following.
Validation Procedure for Viscoplasticity Model with Isotropic and Kinematic Hardening Behavior
The Transitional Markov Chain Monte Carlo method and Gauss-Markov-Kalman filter using the polynomial chaos expansion are applied on a viscoplastic model considering both the isotropic and kinematic hardening behavior.
The parameters which are set as the uncertain parameters are bulk modulus (κ), shear modulus (G), the isotropic hardening coefficient (bR), the kinematic hardening coefficient (bχ) and the yield stress (σy). Hence the vector of unknown parameters is q=
κ(ω), G(ω), bR(ω), bχ(ω), σy(ω).
A preliminary study in the following of our previous studies [88–90] is on a regular cube, modeled with one 8 node element. The minimal number of freedoms that have to be constrained is six and
many combinations are possible. As shown in Figure2, all three degrees of freedom at point B are fixed. This prevents all rigid body translations, and leaves three rotations to be taken care of. The x displacement component at point A is constrained to prevent rotation about z, and the z component is fixed at point C to prevent rotation about y. The y component is constrained at point D to prevent rotation about x [91].
Figure 2. Boundary condition considered.
The normal tractions, which is a Neumann boundary condition are applied cyclically in x, y and z directions on front and back faces and the magnitude of tractions in all directions are shown in Figure3
where green, red and blue colors represent the stress values in x, y and z directions respectively.
0 5 10 15 −1.5 −1 −0.5 0 0.5 1 1.5x 10 9 Time [s] Force [N] sigma(0) sigma(1) sigma(2)
Figure 3. Decomposed applied force at point E according to time.
By considering the parameters listed in Table2, for the top right corner node on back face, point E, as shown in Figure2, the related displacement graph is obtained as shown in Figure4where green, red and blue colors represent the displacement of point E in x, y and z directions respectively.
0 5 10 15 −1.5 −1 −0.5 0 0.5 1 1.5 Time [s] Displacement [mm] u(0) u(1) u(2)
Table 2.The model parameters.
κ G σy n k bR HR bχ Hχ
1.66×109 7.69×108 1.7×108 1 1.5×108 50 2.75×108 50 2.75×108
The displacements of point E in x, y and z directions are noted as the virtual data in this case. Applying both the stochastic identifications, which are fully discussed in Sections3.1and3.2, i.e., Transitional Markov Chain monte Carlo method and history matching Gauss-Markov-Kalman filter and sequential Gauss-Markov-Kalman filter approach, the probability density functions of prior and posterior of the identified parameters can be seen and compared in Figure5. It should be pointed out that in the history matching updating the whole time interval is considered and update is done only once by comparing the predicted value with the recorded displacement history. On the other hand in the sequential updating the process of updating is done several times.
Summarizing the results, the true values and the mean and variance of the estimated parameters for each Bayesian method are compared in Table3.
Table 3.The identified model parameters.
Param. qtrue qmTMCMC qstdTMCMC qGMKFm qstdGMKF qmSGMKF qstdSGMKF κ 1.66×109 1.71×109 1.33×107 1.66×109 2.28×107 1.66×109 1.13×107 G 7.69×108 7.66×108 2.24×106 7.68×108 9.09×106 7.68×108 3.47×106 bR 50 53.00 1.83 54.78 5.16 52.36 3.71 bχ 50 49.30 1.32 54.58 5.15 52.04 3.01 σy 1.7×108 1.67×108 1.37×106 1.67×108 3.48×106 1.69×108 1.35×106 1.2 1.4 1.6 1.8 2 2.2 2.4 x 109 0 0.5 1 1.5 2 2.5 3 3.5 4x 10 −8 Value κ −PDF κ−prior−TMCMC κ−posterior−TMCMC κ−prior−GMKF κ−posterior−GMKF κ−posterior−SGMKF κ−True 7 7.5 8 8.5 9 9.5 x 108 0 0.5 1 1.5 2 2.5x 10 −7 Value G−PDF G−prior−TMCMC G−posterior−TMCMC G−prior−GMKF G−posterior−GMKF G−posterior−SGMKF G−True 1.4 1.6 1.8 2 2.2 2.4 2.6 x 108 0 0.5 1 1.5 2 2.5 3 3.5 4x 10 −7 Value σy −PDF σy−prior−TMCMC σy−posterior−TMCMC σy−prior−GMKF σy−posterior−GMKF σy−posterior−SGMKF σy−True 30 40 50 60 70 80 90 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Value bR −PDF bR−prior−TMCMC bR−posterior−TMCMC bR−prior−GMKF bR−posterior−GMKF bR−posterior−SGMKF bR−True 30 40 50 60 70 80 90 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Value bχ −PDF bχ−prior−TMCMC bχ−posterior−TMCMC bχ−prior−GMKF bχ−posterior−GMKF bχ−posterior−SGMKF bχ−True
Figure 5. PDF of identified parameters.
5. Discussion and Comparison
By comparing the results from the two main approaches, it can be concluded Transitional Markov Chain Monte Carlo and Gauss-Markov-Kalman filter method provide accurate result and identify the known parameters for the viscoplasticity model with isotropic and kinematic hardening behavior except in the history matching update approach as the Gauss-Markov-Kalman filter method for the hardening parameters are not as narrow as the rest of parameter’s distributions as shown in Figure5. This is because the hardening parameters have no direct and strong influence on the measured
displacement and so the uncertainties are not reduced good enough than rest of the parameters as there is no enough information. As another reason it can be mentioned that updating is done only once when history matching Gauss-Markov-Kalman filter is applied and hence the posterior distributions of hardening parameters are not as narrow as when sequential Gauss-Markov-Kalman filter is employed.
In addition, as the computations are evaluated from the initial time step after each update in the sequential Gauss-Markov-Kalman filter, more information is recorded in total. Therefore, better identification and reduction of uncertainty can be obtained in comparison with history matching Gauss-Markov-Kalman filter.
There is not much difference between the standard deviation of the uncertain parameters between Transitional Markov Chain Monte Carlo approach and Gauss-Markov-Kalman filter approach for sequential updating of viscoplasticity with isotropic and kinematic hardening behavior. It should also be noted that the Transitional Markov Chain Monte Carlo method computational time is significantly higher than the other approach. The computational time is more in the Gauss-Markov-Kalman filter approach as multiple updating is involved in the sequential updating of Gauss-Markov-Kalman filter approach.
As Transitional Markov Chain Monte Carlo approach applied on viscoplasticity model with isotropic and kinematic hardening is computationally too much expensive, Gauss-Markov-Kalman filter method is only considered for the viscoplastic-damage model in Section5.1.
5.1. Validation Procedure on Viscoplastic-Damage Model with Isotropic and Kinematic Hardening
In this section the capability of only one of the Bayesian approaches discussed in the previous sections is studied. Gauss-Markov-Kalman filter using the polynomial chaos expansion is applied on a viscoplastic-damage model with both the isotropic and kinematic hardening behavior.
The parameters which are set as the uncertain parameters are bulk modulus (κ), shear modulus (G), the isotropic hardening coefficient (bR), the kinematic hardening coefficient (bχ) and the yield stress (σy) as well as damage parameters ((c1), (c2) and (c3)). Hence the vector of unknown
parameters is q=
κ(ω), G(ω), σy(ω), bR(ω), bχ(ω), c1(ω), c2(ω), c3(ω).
Similar to the previous case, preliminary study is on a regular cube, modeled with one 8 node element as shown in Figure2. Similarly the same Dirichlet boundary condition is considered. Also the same tractions which is a Neumann boundary condition are applied cyclically in x, y and z directions on front and back faces and the magnitude of tractions in all directions are shown in Figure6where green, red and blue colors represent the stress values in x, y and z directions respectively.
0 2 4 6 8 10 −1500 −1000 −500 0 500 1000 1500 Time [s] Force [N] sigma(0) sigma(1) sigma(2)
Figure 6. Decomposed applied force at point E according to time.
By considering the parameters listed in Table4, for the top right corner node on back face, point E, as shown in Figure2, the corresponding displacement graph is obtained as shown in Figure7where green, red and blue colors represent the displacement of point E in x, y and z directions respectively.
Table 4.The model parameters. κ G σy n k bR HR bχ Hχ c1 c2 c3 1.66×105 7.69×104 266 1 23,500 298.6 117.2 100 150 7 8 −80 0 2 4 6 8 10 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 Time [s] Displacement [mm] u(0) u(1) u(2)
Figure 7. Displacement of point E in x, y and z directions according to time.
Again the displacements of point E in x, y and z directions are noted as the virtual data in this case. Applying the Gauss-Markov-Kalman filter approach, the probability density functions of prior and posterior of the identified parameters can be seen and compared in Figure8.
1.5 1.6 1.7 1.8 1.9 2 2.1 x 105 0 0.2 0.4 0.6 0.8 1x 10 −3 Value κ −PDF κ−prior κ−posterior−GMKF κ−posterior−SGMKF κ−True 7 8 9 10 x 104 0 0.2 0.4 0.6 0.8 1 1.2x 10 −3 Value G−PDF G−prior G−posterior−GMKF G−posterior−SGMKF G−True 240 260 280 300 320 340 0 0.1 0.2 0.3 0.4 0.5 Value σy −PDF σy−prior σy−posterior−GMKF σy−posterior−SGMKF σy−True 2000 250 300 350 400 450 0.02 0.04 0.06 0.08 0.1 Value bR −PDF bR−prior bR−posterior−GMKF bR−posterior−SGMKF bR−True 60 80 100 120 140 160 180 0 0.05 0.1 0.15 0.2 0.25 Value bχ −PDF bχ−prior bχ−posterior−GMKF bχ−posterior−SGMKF bχ−True 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Value c1 −PDF c1−prior c1−posterior−GMKF c1−posterior−SGMKF c1−True 4 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Value c2 −PDF c2−prior c2−posterior−GMKF c2−posterior−SGMKF c2−True −1600 −140 −120 −100 −80 −60 −40 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Value c3 −PDF c3−prior c3−posterior−GMKF c3−posterior−SGMKF c3−True
Figure 8. PDF of identified parameters.
From the sharpness of the posterior PDF of κ, G and σy, it can be concluded that enough
information from virtual data is received, and updating the parameters considering their uncertainty is done much easier than the hardening and damage parameters. One reason that can be mentioned
is that the process is not always in the states that hardening and damage equations are involved. Therefore less information from the whole simulation can be analyzed to estimate the hardening parameters and update their parameters’ uncertainties. As the evaluation of hardening and local damage are seen in Figure9, only in the time intervals [0.8 2.0], [3.5 5.0] and [6.6 8.2], and [0.8 2.0] and [6.6 8.2] hardening equations and damage equations are involved respectively. Therefore hardening and damage parameters are updated, but not as much as the rest of the parameters.
0 2 4 6 8 10 0 20 40 60 80 100 120 140 Time [s] Value [−] Isotropic Kinematic 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 Time [s] Value [−] Damage
Figure 9. Isotropic and kinematic hardenings- and local damage evaluation according to time [s]. Summarizing the results, the true values and the mean and variance of the estimated parameters are compared in Table5.
Table 5.The identified model parameters.
Param. qtrue qmGMKF qstdGMKF qmSGMKF qstdSGMKF κ 1.66×105 1.66×105 2.03×103 1.66×105 937.87 G 7.69×104 7.70×104 710.34 7.70×104 430.05 σy 266 264.75 1.81 265.1 1.07 bR 298.6 304.75 29.02 300.35 7.06 bχ 100 103.12 11.18 102.98 2.77 c1 7 7.07 0.82 6.84 0.65 c2 8 7.82 0.70 8.13 0.57 c3 −80 −86.29 10.29 −79.98 3.34 6. Conclusions
The updating is possible by Gauss-Markov-Kalman filter approach for the viscoplastic-damage model with isotropic and kinematic hardening as the updating is performed according to the history matching or sequentially updating method. However it should be pointed out that for the parameters such as hardening and damage parameters, which do not have much strong influence on the measured displacement than the rest of parameters, the identification can not be done well as compared to the rest of parameters. Hence a better identification is achieved when the more information is available from the measurement.
Moreover the uncertainty of the parameters is reduced in the such a way that the posterior distribution functions of the hardening and damage parameters are not as narrow as the rest parameters for the history matching and sequential updating methods.
The sequential updating needs more computational time as the updating process is performed several times than the history updating approach. By considering the fact that the computations are done from the initial time step after each update which leads to observe more information as well. On the other hand, the identification and reduction of the parameter uncertainty as inferred from the update of the parameters is more accurate and acceptable for the sequential updating method than the history matching update as it is seen from the posterior densities of hardening and damage parameters by both approaches shown in Figure8.
7. Summary
Using the stochastic methods explained in Section3to identify the model parameters of the Chaboche damaged model indicates that it is possible to identify the model parameters using Transitional Markov Chain Monte Carlo method and Gauss-Markov Kalman filter approach by means of history updating and also sequentially although the mentioned methods provide different results and the conclusions are discussed below.
The computational time of the Transitional Markov Chain Monte Carlo method on the mentioned model is very high and this results in no sense of applying this method in the economical point of view on the complicated material models. The sequential Gauss-Markov-Kalman filter approach for both variety of the considered models is not as fast as the history matching approach as updating is done multiple times. But still the computation time needed is reasonable and acceptable as a stochastic approach to identify the parameters of a PDE system.
A considerable difference in results is observed between the updated random variables of the uncertain parameters for the considered models which are determined by employing the Gauss-Markov-Kalman filter approach. This is because of the fact that the hardening and damage parameters do not have a strong and direct influence as like the bulk modulus, shear modulus and the yield stress which have a dominant influence on the considered measured displacement output data. The Gauss-Markov-Kalman filter approach in a history matching update approach is not an optimized method to identify the parameters of the viscoplasticity which includes hardening model and viscoplastic-damage model. This is because, the hardening and damage parameters are not identified well and their uncertainties are not properly reduced as updating is done only once. Further, it is also to be noted that the number of observed states where the bulk modulus, shear modulus and yield stress are involved and identifiable, i.e., the number of states where the elasticity and plasticity part are involved and identifiable are more than the number of states where the hardening and damage parameters are involved and identifiable i.e., only plasticity part.
Based on the reliability, accuracy and computational time, the most suitable method among the employed methods is the sequential Gauss-Markov-Kalman filter approach for the viscoplastic model with isotropic and kinematic hardening and also for the viscoplastic-damage model enriched with hardening behavior. The parameters such as hardening parameters, damage parameters, bulk modulus, shear modulus and yield stress are well updated and posterior distributions representing their uncertainties are much narrow for these parameters. Using sequential Gauss-Markov-Kalman filter approach the parameters should be updated at proper time intervals to save time and to reduce cost. Lesser number of updates results in lesser computation time. Hence particular times should be selected to identify the parameters as discussed in Section4where the hardening and damage parameters are changing continuously. Considering this fact results in a better parameter identification and better reduction of parameter uncertainties in a very less time.
It should be noted that the damage evolution has been restricted in such a way that non-local damage should not exceed certain value, i.e., only minor damage is considered for the Gauss-Markov-Kalman filter approach on a viscoplastic-damage model to identify its parameters. Obviously the damage parameters are identified better by using the Gauss-Markov-Kalman filter approach when an immense damage is considered, i.e., the measured displacement data is influenced more by a bigger damage and results in better identification of damage parameters from the provided data. Considering the mentioned fact, this study proves that identification of the model and damage parameters are possible using Bayesian methods before the specimen is really damaged and completely collapsed. Such a Bayesian approach can be recommended to be applied for mechanical structure health monitoring [92,93].
Author Contributions: Conceptualization, formal analysis, investigation, methodology, software, validation, visualization, writing—original draft, writing—review and editing have been done by E.A., H.G.M. and D.D. were in charge of supervison, methodolgy, conceptualization and funding acquisition. B.R. and S.R. have helped E.A. in validation and methodology. All authors have read and agreed to the published version of the manuscript.
Funding:This work is partially funded by the DFG through GRK 2075. Conflicts of Interest:The authors declare no conflict of interest.
Appendix A. Elasto-Viscoplastic Constitutive Laws
Neglecting inertia and using infinitesimal strain theory, the mechanics of the investigated structures with elasto-viscoplastic material behavior is described by the
equilibrium − ∇·σ =ρg (A1)
and the strain balance eel+evp =e (A2)
see Bonet and Wood [94]. The Cauchy stress σ is caused by the body force ρg specified by density ρ and gravity g and the boundary conditions
u= ¯u (A3)
σ·n=¯t (A4)
on the Dirichlet respectively Neumann part. Here the impressed displacement ¯u and the impacting surface tension ¯t are marked with a bar, the used symbol for a boundary condition. In Equation (A4) the Cauchy stress is mapped to the boundary by the normal vector n pointing outwards with respect to the structural body. Next to the equilibrium (A1) the strain balance (A2) changes into a differential equation as well when the kinematics of the structure is taken into account. The kinematics is described by the Cauchy strain tensor
e= 1 2
∇u+ (∇u)T (A5)
given by the gradient of the displacement field. In the following, the constitutive equations for the elastic and viscoplastic strains, eelrespectively evp, linking the governing Equations (A1) and (A2) are
specified in detail.
For mild steel under cyclic loading beyond the yield limit taking isotropic-kinematic hardening into account as well as viscoplastic damaging the material behavior is described by a modified Chaboche model [95] introduced by Kowalsky et al. [74]. Here, damage is described by continuum damage mechanics with evolution equation for isotropic ductile damage D, defined later, spatially distributed with the differential equation
D=D¯ −l2c∇2D¯ (A6)
depending on the internal length lc as model parameter. Equation (A6) introduces the non-local
damage variable ¯D, marked with a bar as well and used in the concept of effective strains and stresses, see Chaboche [96]. By Equation (A6) it is assumed that damage occurs locally but acts non-locally. It is supposed that only the stress is affected by damage while the strain remains unaffected. The elastic behavior is described by the potential
φel(eel) =
λ
2 (tr(eel))
2+
µ eel: eel (A7)
of a St. Venant-Kirchhoff material. The second Lamé-coefficients µ is alternatively named as shear modulus G in the following. Together with the first Lamé-coefficient, the elastic behavior of the material is further described by the bulk modulus κ=λ+23µ, used from this point onwards. With the
tensor of elasticity
C= ∂ 2φ
el
the rate formulation of the constitutive equation for the elastic part of the effective strains reads: ˙˜eel=C−1 : ˙˜σ= 1 1−D¯ C−1 : ˙σ+ ˙¯ D (1−D¯)2 C −1: σ. (A9)
Appendix B. Gauss-Markov-Kalman Filter
Appendix B.1. Conditional Expectation
The conditional expectation is defined on the Hilbert space where the random variables are considered with finite variance as shown in Equation (A10) where the considered sub-σ-algebra B is a subset of the underlying σ-algebra A, i.e., B ⊂ A. It should be noted that the σ-algebra is basically representing the collection of subsets ofΩ on which statements about their probability can be made. Therefore it has a continuous orthogonal projection PB : S → SBfrom the whole
σ-algebra to sub-σ-algebra B that has the collection of subsets ofΩ based on which their probability is
determined [69,97].
SB:=L2(Ω, B,P):= {r :Ω→ R : r measurable w.r.t. B ,E (|r|2) <∞} ⊂ S (A10)
Thus the conditional expectation (CE) of a random variable r∈ S with respect to a sub-σ-algebra Bcan be defined by the orthogonal projection as shown in Equation (A11).
E(r|B):=PB(r) ∈ SB (A11)
The conditional expectation minimizes the squared error as it is an orthogonal expectation as shown in Equation (A12).
E(|r− E(r|B)|2) =min{E(|r−˜r|2) : ˜r∈ SB} (A12)
Equation (A12) leads to the orthogonality relation∀˜r∈ SB : E(˜r(r− E(r|B))) =0. Also a form
of Pythagoras theorem [98] can be considered as shown in Equation (A13).
E(|r|2) = E(|r− E(r|B)|2) + E(|E(r|B)|2) (A13)
Therefore it can be concluded that conditional expectation is a form of a minimum mean square error estimator. The conditional probability, for instance, the posterior in Bayes’s theorem can be characterized by the conditional expectation [69,99]. For instance the conditional probability for A⊂Ω, A∈Bis obtained by Equation (A14) where the random variable χAbecomes unity if ω∈ A and it
vanishes otherwise.
P(A|B):= E(χA|B) (A14)
Based on Equation (A14), if the conditional expectationE(χA|B)is known then everything
about the conditional probabilityP(A|B) is known and thus the posterior density is defined by
this. If the distribution of a random variable representing the prior density is characterized by the prior characteristic function then the conditional characteristic function is determined by using the conditional expectation and this characterizes the conditional distribution. If the prior probability was the distribution of some random variable r, then it is completely characterized by the prior characteristic function, i.e., in the sense of probability theory, it can be expressed as φr(s):= E(exp(irs))[70].
To determine the conditional characteristic function φr|B(s):= E(exp(irs)|B), conditional expectation is used instead of the unconditional expectation and this completely characterizes the conditional distribution.
For an instance, if a random variable y is the observation and the sub-σ-algebra B is generated from the observation y resulting in B=σ(y), the information about the observation can be obtained
only on B=σ(y)which are some subsets ofΩ [69]. According to the Doob-Dynkin lemma [100,101],
the Equation (A15) represents the subspaceSσ(y)as the functions of observation.
Sσ(y):= {r∈ S : r(ω) =φ(y(ω)), φ measurable} ⊂ S (A15)
It can be inferred from the Equation (A15) that an observation is a function of the observation and where the information from the measurement is lying is represented by the subspaceSσ(y)⊂ S[69].
The conditional expectation E(r|σ(y)) = E(r|y) and the conditional probability P(A|σ(y)) = P(A|y)are random variables as y is a random variable. If a fixed value ˆy∈ Yis observed
for the random variable y, i.e., an observation is established then the posterior expectation is just a numberE(r|ˆy) ∈ Rand the posterior probability isP(A|ˆy) = E(χA|ˆy). Accordingly, Equation (A16)
can be determined for some function φr from the Equation (A15) which means for each random
variable r it is a possibly different function.
E(r|y) =φr(y) and E(r|ˆy) =φr(ˆy) (A16)
Considering Bayes’s theorem [69,70,101–104] , it can be concluded that if it is possible to compute the conditional expectation with respect to an observation y, then the conditional probability with respect to observation ˆy can also be computed and this leads to determine the posterior probability as described in Bayes’s theorem. Thus, the Bayesian estimation can be done by using the concept of conditional expectation [69–71].
Appendix B.2. Constructing a Posterior Random Variable
Equation (A17), which is the probabilistic model of the observation is obtained by considering the random variables and the observation equation of the model as discussed in Equation (10) in Section2
where the mapping Y is observed. This mapping is on the Hilbert space of the random variables with finite variance.
ˆy=y+e=Y(q) +e (A17)
Appendix B.3. Updating Random Variables
Let’s assume a random variable qn∈Q, the next state ˆqn+1∈Q and the measurement yn+1∈Y
as time tngoes to tn+1can be predicted. The conditional expectation of the measurement prediction
yn+1calculated based on Equation (A17) is shown in Equation (A18) and the posterior expectation
operator is computed by considering the actual observation ˆyn+1as shown in Equation (A19).
E(Ψ(qn+1)|σ(yn+1)) =φΨ(yn+1) (A18) E(Ψ(qn+1)|ˆyn+1) =φΨ(ˆyn+1) (A19)
As the conditional expectation of the posterior is known, all the information of the posterior probability is available [69–71].
It is to be noted that the Equation (A17) requires every time new random variable qn+2to be
computed from tn+1to tn+2and this new random variable has the posterior distribution given by
the mappings φΨ(ˆyn+1)as shown in Equation (A19). Although there are many random variables which have this posterior density, only one particular distribution should be chosen via the method as discussed in this section.
To simplify the notations, the forecast random variable is considered as qf = ˆqn+1 and the
forecast measurement is considered as yf = yn+1. The measurement is represented as ˆy = ˆyn+1.
Therefore the update of the forecast random variable qf, also called as the assimilated random
an innovation map which plays the role of a transformation so-called filter i.e., update by filtering the measurement ˆy [70,71].
qa=B(qf, yf, ˆy) =qf +Ξ(qf, yf, ˆy) (A20)
Appendix B.4. Correcting the Mean
To correct the mean value of the new random variable ¯qa= E(qa|ˆy), it should be considered that E(qa|σ(yf)) =φqf(yf):=φq(yf)is an orthogonal projection Pσ(yf)(qf)fromQ= Q ⊗ SontoQ∞:=
Q ⊗ S∞, whereS∞ := Sσ(y) = L2(Ω, σ(yf),P)by considering the Equations (A11) and (A19) [69].
Therefore the orthogonal decomposition can be shown as in Equation (A21). Q= Q ⊗ S =Q∞⊕Q⊥
∞= (Q ⊗ S∞) ⊕ (Q ⊗ S∞⊥) (A21)
qf =Pσ(yf)(qf) + (IQ−Pσ(yf))(qf) =φq(yf) + (qf −φq(yf)) (A22)
Equation (A23) can be determined as the conditional expectation of the second term in Equation (A22) is zero as this term is representing the projection part of the decomposition, i.e.,E(qf −φq(yf)|σ(yf)) = Pσ(yf)(IQ−Pσ(yf))(qf) = 0 [69–71]. It can be also explained in this
way that when the measurement is obtained, the Equation (A22) is changed by fixing the component
φq(yf) ∈Q∞and leaving the orthogonal rest unchanged.
qa,1=φq(ˆy) + (qf −φq(yf)) =qf + (φq(ˆy) −φq(yf)) (A23)
The posterior mean as seen in Equation (A24) can be obtained from Equation (A23).
¯qa,1= E(qa,1|ˆy) =φq(ˆy) = E(qa|ˆy) (A24)
Appendix B.5. Building the Filter
The algorithm to update the random variable with a map g : Y → Qto approximate φq is obtained by inserting the Equation (A17) into Equation (A23) to obtain the Equation (A25).
qn+1=qn+ (g(ˆyn+1) −g(Y(q) +e)) =qn−g(Y(q) +e) +g(ˆyn+1) (A25)
It is inferred from the Equation (A25) that the model Equation (A17) is corrected by an innovation term and the Equation (A25) can be named as the filter equation for identifying the extended uncertain parameters of Equation (A17) which is an unbiased filter with the minimum mean square error estimate represented by φ(ˆy).
The map φΨ is defined by Equation (A26). It is obtained by introducing v ranging over all measurable maps v : Y → Q and by using the Equation (A12) representing the combination of minimization property and the Equation (A13) representing the Doob-Dynkin lemma [100,101].
||Ψ(q) −φΨ(y)||2Q=min v
||Ψ(q) −v(y)||2Q= min
z∈Q∞
||Ψ(q) −z||2Q (A26)
AsQσ(y) = Q∞isL-closed, it can be concluded that∀z ∈ Q∞ : E(z⊗ (Ψ(q) −φΨ(y)))[105]. Hence the random variable (Ψ(q) −v(y)) is orthogonal in the L-invariant sense to all random
variables z ∈ Q∞. In other words, the correlation operator of (Ψ(q) −v(y)) is zero [69–71].
It should also be noted that the measurement operator evaluating y need not necessarily be linear in q and hence the optimal map φq(y)also need not necessarily be linear in y. However the conditional expectationE(q|y) =Pσ(y)(q)is basically an orthogonal projection.
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